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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 57: Manifolds and cell complexes

Introduction

Manifolds are spaces like the sphere which look locally like Euclidean space. In particular, these are the spaces in which we can discuss (locally-)linear maps, and the spaces in which to discuss smoothness. They include familiar surfaces. Cell complexes are spaces made of pieces which are part of Euclidean space, generalizing polyhedra. These types of spaces admit very precise answers to questions about existence of maps and embeddings; they are particularly amenable to calculations in algebraic topology; they allow a careful distinction of various notions of equivalence. These are the most classic spaces on which groups of transformations act. This is also the setting for knot theory.

History

Thorough, excellent coverage is provided by Dieudonné, Jean, "A history of algebraic and differential topology. 1900--1960", Birkhäuser Boston, Inc., Boston, MA, 1989, 648 pp. ISBN 0-8176-3388-X MR90g:01029. He has a similar (shorter!) survey done a little later (see MR95k:55001)

Perhaps it is easiest to use classic literature to understand differential topology: Flatland; here are two Backup sites and the home page for Project Gutenberg.

Applications and related fields

There are two other topology pages:

Algebraic topology -- definitions and computations of fundamental groups, homotopy groups, homology and cohomology. This includes Homotopy theory -- studies of spaces in the homotopy category (but without applications to questions on algebraic invariants)

General topology is for spaces without the local Euclidean nature of the spaces in this section. There is however some semblance to the present topic among metric spaces.

Simplicial complexes are essentially polyhedra.

Problems specific to Euclidean space may be treated in the geometry pages

Questions regarding, say, smoothness on balls and spheres are in the FAQ for spheres.

For complex manifolds, See 32C10

Geometric topology is a natural language in which to study families of motions; applications include some topics in mechanics of moving particles and systems.

Subfields

• 57M: Low-dimensional topology, including Knot theory
• 57N: Topological manifolds
• 57P: Generalized manifolds, see also 18F15
• 57Q: PL-topology (Triangulation is part of 57R)
• 57R: Differential topology For foundational questions of differentiable manifolds, see 58AXX; for infinite-dimensional manifolds, See 58BXX
• 57S: Topological transformation groups, see also 20F34, 22-XX, 54H15, 58D05
• 57T: Homology and homotopy of topological groups and related structures

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Bourbaki, N., "Variétés différentielles et analytiques" Hermann, Paris 1971 99 pp.

Fox, R. H.: "A quick trip through knot theory", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 120--167; Prentice-Hall, Englewood Cliffs, N.J. 1962 MR25#3522

There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period.

Cannon, J. W.: "The recognition problem: what is a topological manifold?", Bull. Amer. Math. Soc. 84 (1978), no. 5, 832--866. MR58#13043

Software and tables

• Snappea is a collection of interconnected programs for analyzing hyperbolic manifolds. In particular, it analyzes the hyperbolic structure of knot complements.

Other web sites with this focus

• Algebraic Topology Discussion List
• UTK archive page
• Here are the AMS and Goettingen resource pages for area 57.

Selected topics at this site

• Using the language of maps between manifolds to discern whether or not a function of several variables can be "simplified".
• How many homeomorphisms of an interval are there, having order 2?
• Deriving the equations of a torus.
• Parameterization of Klein bottle, Möbius strip
• What is the dimension of a manifold (e.g. what dimensional creatures "live on" a Klein bottle?)
• Why do you get linked pieces if you cut a Möbius strip in half?
• Cataloguing automorphisms of a surface
• If the 2-sphere is written as the union of two compact pieces K and L having finitely many components, then (K intersect L) has finitely many components.
• Introduction to Dessins des enfants
• The tangent bundle and tensor bundles over manifolds
• Orientable 3-manifolds are parallelizable
• Which dimensional spheres are parallelizable?
• Parallelizability in smooth and analytic categories
• Can you comb the hair on a sphere? (no)
• A vector field tangent to an odd-dimensional sphere must vanish somewhere.
• Number, construction of independent vector fields on an n-sphere
• When does a manifold admit a smooth nowhere-vanishing vector field?
• The canonical line bundle over projective space
• Description and application of the "Plate trick", a parlor trick giving a concrete example of a homotopy class of order 2!
• The plate trick -- a physical manifestations of a path of order 2 in pi_1(SO*(3)).
• How is it that the "plate trick" demonstrates pi_1(SO_3) is Z/2Z ?
• Some puzzles testing linking and homotopy intuition.
• Difference between PL and differentiable manifolds.
• Computing locations in a simplicial complex using barycentric coordinates
• Algorithm to determine whether a 3-complex is homeomorphic to S^3
• Kuratowski-like conditions for embeddability of a 2-simplex into R^3?
• Is a cover of a cover again a cover? (no)
• An application of covering spaces to complex analysis
• Global Complex Analysis is differential topology; low-dim manifolds which are groups
• Example of use of flag manifolds and counting stabilizers to enumerate orbits of subspaces
• Grassmannians are topological spaces which enumerate subspaces of a given dimension. Here is a long exchange with a person seeking to randomly select subspaces in a uniform way.
• Using Grassmannians to clarify the concept of sets of circles in the 3-sphere.
• An article describing one particular Grassmannian space.
• John Baez describes framed embeddings
• Elementary proofs of the Borsuk-Ulam theorem
• If X is a fundamental domain for the usual action of Z x Z on the plane, how do we determine in which translate of X a point of the plane lies?
• Dan Asimov asks about "hooples"
• Another proof that mathematicians have a language all their own :-)